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Question
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
Options
True
False
Solution
This statement is False.
Explanation:
Let Sn = an2 + bn + c ...(Quadratic expression)
S1 = a + b + c
∴ a1 = a + b + c
S2 = 4a + 2b + c
a2 = S2 – S1
= (4a + 2b + c) – (a + b + c)
= 3a + b
S3 = 9a + 3b + c
⇒ a3 = S3 – S2
= (9a + 3b + c) – (4a + 2b + c)
= 5a + b
Common difference d = a2 – a1
= (3a + b) – (a + b + c)
= 2a – c
and d = a3 – a2 = (5a + b) – (3a + b) = 2a
Here, we observe that a2 – a1 ≠ a3 – a2
So it does not represent an A.P
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